Explicit Methods for Rational Points on Curves (07w5063)

Organizers

Objectives

The question of finding the rational solutions to polynomial equations is a natural one. Arguably, it is one of the oldest problems studied by mathematicians. It is also a problem that arises naturally in several other disciplines. Currently, there are many interesting developments for this field.

On one hand, there is an increased theoretical understanding of the mathematical constructions that govern the existence of rational points on curves. Work that links Mordell-Weil sieving, Brauer-Manin obstructions and descent obstructions bears evidence to that.

On the other hand, increases in computational power and symbolic computational methods, together with large amounts of research, are beginning to make it possible to compute the relevant information in many explicit situations needed to determine the rational points on curves.

Both the theoretical developments and the practical advances benefit from each other's progress. Because solving even the simplest-looking problems in the arithmetic of curves often requires techniques from other parts of number theory and algebraic geometry, our participant list includes experts active in several subfields:

* p-adic analysis

* Chabauty methods

* Descent

* L-series

* computational algebraic number theory

* constructive class field theory

One very specific goal of the workshop is to obtain a deeper understanding of Minhyong Kim's new approach to "non-abelian Chabauty" (http://front.math.ucdavis.edu/math.NT/0409456). There are many questions here: How does this approach relate to applying the original Chabauty method to unramified covers? Can it be made practical? Our workshop would be the first to bring Kim's ideas into contact with the leaders in computational number theory. Kim has already agreed to participate, and we expect that by having him present a series of lectures, his work will be made accessible, despite the depth of the mathematics involved.

Because we intend to have several young participants, and because we will have participants coming from different subfields of arithmetic geometry, we plan to begin the workshop with a few survey lectures, focusing on topics such as Chabauty methods and descent obstructions. These will help to align the participants towards the special goal mentioned above, as well as provide young talent with a quick introduction into the field.